AIRSP OF COMMUTING ISOMETRIES - Ijay/papers/Pure Pair Final Studia Math 2018.… · commuting isometries. Finally, we study the defect operators of pairs of commuting isometries

PAIRS OF COMMUTING ISOMETRIES - I

AMIT MAJI, JAYDEB SARKAR, AND SANKAR T. R.

Abstract. We present an explicit version of Berger, Coburn and Lebow'sclassi�cation result for pure pairs of commuting isometries in the senseof an explicit recipe for constructing pairs of commuting isometric mul-tipliers with precise coe�cients. We describe a complete set of (joint)unitary invariants and compare the Berger, Coburn and Lebow's repre-sentations with other natural analytic representations of pure pairs ofcommuting isometries. Finally, we study the defect operators of pairs ofcommuting isometries.

1. Introduction

A very general and fundamental problem in the theory of bounded linear

operators on Hilbert spaces is to �nd classi�cations and representations of

commuting families of isometries.

In the case of single isometries this question has a complete and ex-

plicit answer: If V is an isometry on a Hilbert space H, then there exist

a Hilbert space Hu and a unitary operator U on Hu such that V on H

and

[S ⊗ IW 0

0 U

]on (l2(Z+) ⊗ W) ⊕ Hu are unitarily equivalent, where

W = kerV ∗ is the wandering subspace for V and S is the forward shift

operator on l2(Z+) [H]. This fundamental result is due to J. von Neumann

[VN] and H. Wold [W] (see Theorem 2.1 for more details).

The case of pairs (and n-tuples) of commuting isometries is more subtle,

and is directly related to the commutant lifting theorem [FF] (in terms of

an explicit, and then unique solution), invariant subspace problem [HH] and

representations of contractions on Hilbert spaces in function Hilbert spaces

[NF]. For instance:

(a) Let S be a closed joint (Mz1 ,Mz2)-invariant subspace of the Hardy space

H2(D2). Then (Mz1|S ,Mz2|S) on S is a pure (see Section 3) pair of commut-

ing isometries. Classi�cation of such pairs of isometries is largely unknown

(see Rudin [R]).

2010 Mathematics Subject Classi�cation. Primary 47A05, 47A13, 47A20, 47A45,47A65; Secondary 46E22, 46E40.

Key words and phrases. Isometries, commuting pairs, commutators, multipliers,Hardy space, defect operators.

1

2 A. MAJI, J. SARKAR, AND SANKAR T. R.

(b) Let T be a contraction on a Hilbert space H. Then there exists a pair

of commuting isometries (V1, V2) on a Hilbert space K such that T and

PkerV ∗2V1|kerV ∗2

are unitarily equivalent (see Bercovici, Douglas and Foias

[BDF]).

(c) The celebrated Ando dilation theorem (see Ando [A]) states that a

commuting pair of contractions dilates to a commuting pair of isometries.

Again, the structure of Ando's pairs of commuting isometries is largely

unknown.

The main purpose of this paper is to explore and relate various natural

representations of a large class of pairs of commuting isometries on Hilbert

spaces. The geometry of Hilbert spaces, the classical Wold-von Neumann de-

composition for isometries, the analytic structure of the commutator of the

unilateral shift, and the Berger, Coburn and Lebow [BCL] representations

of pure pairs of commuting isometries are the main guiding principles for

our study. The Berger, Coburn and Lebow theorem states that: Let (V1, V2)

be a pair of commuting isometries on a Hilbert space H, and let V = V1V2

be a shift (or, a pure isometry - see Section 2). Then there exist a Hilbert

space W , an orthogonal projection P and a unitary operator U on W such

that

Φ1(z) = U∗(P + zP⊥) and Φ2(z) = (P⊥ + zP )U (z ∈ D),

are commuting isometric multipliers in H∞B(W)(D), and (V1, V2, V ) on H and

(MΦ1 ,MΦ2 ,Mz) on H2W(D) are unitarily equivalent (see Bercovici, Douglas

and Foias [BDF] for an elegant proof).

Here and further on, given a Hilbert space H and a closed subspace S of

H, PS denotes the orthogonal projection of H onto S. We also set

P⊥S = IH − PS .

In this paper we give a new and more concrete treatment, in the sense of

explicit representations and analytic descriptions, to the structure of pure

pairs of commuting isometries. More speci�cally, we provide an explicit

recipe for constructing the isometric multipliers (Φ1(z),Φ2(z)), and the op-

erators U and P involved in the coe�cients of Φ1 and Φ2 (see Theorems

3.2 and 3.3). Then we compare the Berger, Coburn and Lebow represen-

tations with other possible analytic representations of pairs of commuting

isometries.

In Section 6, which is independent of the remaining part of the paper,

we analyze defect operators for (not necessarily pure) pairs of commuting

isometries. We provide a list of characterizations of pairs of commuting

PAIRS OF COMMUTING ISOMETRIES - I 3

isometries with positive defect operators (see Theorem 6.2). Our results hold

in a more general setting with somewhat simpler proofs (see Theorem 6.5 for

instance) than the one considered by He, Qin and Yang [HQY]. Moreover,

we prove that for a large class of pure pairs of commuting isometries the

defect operator is negative if and only if the defect operator is the zero

operator.

The paper is organized as follows. In Section 2 we review the classical

Wold-von Neumann theorem for isometries and then prove a representation

theorem for commutators of shifts. In Section 3 we discuss some basic rela-

tionships between wandering subspaces for commuting isometries, followed

by a new and explicit proof of the Berger, Coburn and Lebow character-

izations of pure pairs of commuting isometries. Section 4 is devoted to a

short discussion about joint unitary invariants of pure pairs of commuting

isometries. Section 5 ties together the explicit Berger, Coburn and Lebow

representation and other possible analytic representations of a pair of com-

muting isometries. Then, in Section 6, we present a general theory for pairs

of commuting isometries and analyze the defect operators. Concluding re-

marks, future directions and a close connection of our consideration with the

Sz.-Nagy and Foias characteristic functions for contractions are discussed

in Section 7.

2. Wold-von Neumann decomposition and commutators

We begin this section by brie�y recalling the construction of the classical

Wold-von Neumann decomposition of isometric operators on Hilbert spaces.

Here our presentation is more algebraic and geared towards the main theme

of the paper. First, recall that an isometry V on a Hilbert space H is said

to be pure, or a shift, if it has no unitary direct summand, or equivalently,

if limm→∞

V ∗m = 0 in the strong operator topology (see Halmos [H]).

Let V be an isometry on a Hilbert space H, and let W(V ) be the wan-

dering subspace [H] for V , that is,

W(V ) = H VH.

The classical Wold-von Neumann decomposition is as follows:

Theorem 2.1. (Wold-von Neumann decomposition) Let V be an isometry

on a Hilbert space H. Then H decomposes as a direct sum of V -reducing

subspaces Hs(V ) =∞⊕

m=0V mW(V ) and Hu(V ) = HHs(V ) and

V =

[Vs 00 Vu

]∈ B(Hs(V )⊕Hu(V )),


where Vs = V |Hs(V ) is a shift operator and Vu = V |Hu(V ) is a unitary oper-

ator.

We will refer to this decomposition as theWold-von Neumann orthogonal

decomposition of V .

Recall that the Hardy space H2(D) is the Hilbert space of all analytic

functions on the unit disc D with square summable Taylor coe�cients (cf.

[H], [RR]). The Hardy space is also a reproducing kernel Hilbert space cor-

responding to the Szegö kernel

S(z, w) = (1− zw)−1 (z, w ∈ D).

For any Hilbert space E , the E-valued Hardy space with reproducing kernel

D× D→ B(E), (z, w) 7→ S(z, w)IE ,

can canonically be identi�ed with the tensor product Hilbert space H2(D)⊗E . To simplify the notation, we often identify H2(D)⊗E with the E-valuedHardy space H2

E(D). The space of B(E)-valued bounded holomorphic func-

tions on D will be denoted by H∞B(E)(D).

Let MEz denote the multiplication operator by the coordinate function z on

H2E(D), that is

(MEz f)(w) = wf(w) (f ∈ H2

E(D), w ∈ D).

Then MEz is a shift operator and

W(MEz ) = E .

To simplify the notation we often omit the superscript and denote MEz by Mz,

if E is clear from the context.

We now proceed to give an analytic description of the Wold-von Neu-

mann construction.

Let V be an isometry onH, and letH = Hs(V )⊕Hu(V ) be the Wold-von

Neumann orthogonal decomposition of V . De�ne

ΠV : Hs(V )⊕Hu(V )→ H2W(V )(D)⊕Hu(V )

by

ΠV (V mη ⊕ f) = zmη ⊕ f (m ≥ 0, η ∈ W(V ), f ∈ Hu(V )).

Then ΠV is a unitary and

ΠV

[Vs 00 Vu

]=

[MW(V )z 00 Vu

]ΠV .

In particular, if V is a shift, then Hu(V ) = {0} and hence

ΠV V = MW(V )z ΠV .


Therefore, an isometry V onH is a shift operator if and only if V is unitarily

equivalent to MEz on H2

E(D), where dim E = dimW(V ).

In the sequel we denote by (ΠV ,MW(V )z ), or simply by (ΠV ,Mz), the Wold-

von Neumann decomposition of the pure isometry V in the above sense.

Let E be a Hilbert space, and let C be a bounded linear operator on

H2E(D). Then C ∈ {Mz}

′, that is, CMz = MzC, if and only if (cf. [NF])

C = MΘ

for some Θ ∈ H∞B(E)(D) and (MΘf)(w) = Θ(w)f(w) for all f ∈ H2E(D) and

w ∈ D.Now let V be a pure isometry, and let C ∈ {V }′ . Let (ΠV ,Mz) be

the Wold-von Neumann decomposition of V , and let W = W(V ). Since

ΠVCΠ∗V on H2W(D) is the representation of C on H and (ΠVCΠ∗V )Mz =

Mz(ΠVCΠ∗V ), it follows that

ΠVCΠ∗V = MΘ,

for some Θ ∈ H∞B(W)(D). The main result of this section is the following

explicit representation of Θ.

Theorem 2.2. Let V be a pure isometry on H, and let C be a bounded

operator on H. Let (ΠV ,Mz) be the Wold-von Neumann decomposition of

V . Set W =W(V ), M = ΠVCΠ∗V and let

Θ(w) = PW(IH − wV ∗)−1C |W (w ∈ D).

Then

CV = V C,

if and only if Θ ∈ H∞B(W)(D) and

M = MΘ.

Proof. Let h ∈ H. One can express h as h =∞∑

m=0

V mηm, for some ηm ∈ W ,

m ≥ 0 (as H =∞⊕

m=0V mW). Applying PWV

∗l to both sides and using the

fact that W = W(V ) = kerV ∗, we obtain ηl = PWV∗lh for all l ≥ 0. This

implies, for any h ∈ H,

(2.1) h =∞∑

m=0

V mPWV∗mh.


Now let CV = V C. Then there exists a bounded analytic function Θ ∈H∞B(W)(D) such that ΠVCΠ∗V = MΘ. For each w ∈ D and η ∈ W we have

Θ(w)η = (MΘη)(w)

= (ΠVCΠ∗V η)(w)

= (ΠVCη)(w),

as Π∗V η = η. Since in view of (2.1)

Cη =∞∑

m=0

V mPWV∗mCη,

it follows that

Θ(w)η = (ΠV (∞∑

m=0

V mPWV∗mCη))(w)

= (∞∑

m=0

Mmz (PWV

∗mCη))(w)

=∞∑

m=0

wm(PWV∗mCη)

= PW(IH − wV ∗)−1Cη.

Therefore

Θ(w) = PW(IH − wV ∗)−1C|W (w ∈ D),

as required. Finally, since the su�cient part is trivial, the proof is complete.

Note that in the above proof we have used the standard projection for-

mula (see, for example, Rosenblum and Rovnyak [RR]) IH = SOT−∞∑

m=0

V mPWV∗m.

It may also be observed that ‖wV ∗‖ = |w|‖V ‖ < 1 for all w ∈ D, and so

it follows that the function Θ de�ned in Theorem 2.2 is a B(W)-valued

holomorphic function in the unit disc D. However, what is not guaranteedin general here is that the function Θ is in H∞B(W)(D). The above theorem

says that this is so if and only if CV = V C.

3. Berger, Coburn and Lebow representations

This section is devoted to a detailed study of Berger, Coburn and Lebow's

representation of pure pairs of commuting isometries. Our approach is dif-

ferent and yields sharper results, along with new proofs, in terms of explicit

coe�cients of the one variable polynomials associated with the class of pure

pairs of commuting isometries. Before dealing more speci�cally with pure


pairs of commuting isometries we begin with some general observations

about pairs of commuting isometries.

Let (V1, V2) be a pair of commuting isometries on a Hilbert space H. Inthe sequel, we will adopt the following notations:

V = V1V2,

W =W(V ) =W(V1V2) = H V1V2H,and

Wj =W(Vj) = H VjH (j = 1, 2).

A pair of commuting isometries (V1, V2) on H is said to be pure if V is

a pure isometry.

The following useful lemma on wandering subspaces for commuting isome-

tries is simple.

Lemma 3.1. Let (V1, V2) be a pair of commuting isometries on a Hilbert

space H. ThenW =W1 ⊕ V1W2 = V2W1 ⊕W2,

and the operator U on W de�ned by

U(η1 ⊕ V1η2) = V2η1 ⊕ η2,

for η1 ∈ W1 and η2 ∈ W2, is a unitary operator. Moreover,

PWVi = ViPWj(i 6= j).

Proof. The �rst equality follows from

I − V V ∗ = (I − V1V∗

1 )⊕ V1(I − V2V∗

2 )V ∗1 = V2(I − V1V∗

1 )V ∗2 ⊕ (I − V2V∗

2 ).

The second part directly follows from the �rst part, and the last claim

follows from (I − V V ∗)Vi = Vi(I − VjV ∗j ) for all i 6= j. This concludes the

proof of the lemma.

Let (V1, V2) be a pure pair of commuting isometries on a Hilbert space

H, and let (ΠV ,Mz) be the Wold-von Neumann decomposition of V . Since

V Vi = ViV (i = 1, 2),

there exist isometric multipliers (that is, inner functions [NF]) Φ1 and Φ2

in H∞B(W)(D) such that

ΠV Vi = MΦiΠV (i = 1, 2).

In other words, (MΦ1 ,MΦ2) on H2W(D) is the representation of (V1, V2) on

H. Following Berger, Coburn and Lebow [BCL], we say that (MΦ1 ,MΦ2) is

the BCL representation of (V1, V2), or simply the BCL pair corresponding

to (V1, V2).


We now present an explicit description of the BCL pair (MΦ1 ,MΦ2).

Theorem 3.2. Let (V1, V2) be a pure pair of commuting isometries on a

Hilbert space H, and let (MΦ1 ,MΦ2) be the BCL representation of (V1, V2).

Then

Φ1(z) = V1|W2 ⊕ V ∗2 |V2W1z, Φ2(z) = V2|W1 ⊕ V ∗1 |V1W2z,

for all z ∈ D.

Proof. Let η in W = V2W1 ⊕W2, and let w ∈ D. Then there exist η1 ∈ W1

and η2 ∈ W2 such that η = V2η1 ⊕ η2. Then V1η = V η1 + V1η2, and hence

Φ1(w)η = (MΦ1η)(w) = (ΠV V1Π∗V η)(w) = (ΠV V1η)(w) = (ΠV V η1+ΠV V1η2)(w).

This along with the fact that V1η2 ∈ W (see Lemma 3.1) gives

Φ1(w)η = (MzΠV η1 + V1η2)(w)

= (Mzη1 + V1η2)(w)

= wη1 + V1η2

= wV ∗2 η + V1η2,

for all w ∈ D. Therefore

Φ1(z) = V1|W2 ⊕ V ∗2 |V2W1z,

for all z ∈ D, as W2 = Ker(V ∗2 ). The representation of Φ2 follows similarly.

In the following, we present Berger, Coburn and Lebow's version of rep-

resentations of pure pairs of commuting isometries. This yields an explicit

representations of the auxiliary operators U and P (see Section 1). The

proof readily follows from Lemma 3.1 and Theorem 3.2.

Theorem 3.3. Let (V1, V2) be a pure pair of commuting isometries on H.Then the BCL pair (MΦ1 ,MΦ2) corresponding to (V1, V2) is given by

Φ1(z) = U∗(PW2 + zP⊥W2),

and

Φ2(z) = (P⊥W2+ zPW2)U,

where

U =

[V2|W1 0

0 V ∗1 |V1W2

]:W1

⊕V1W2

→V2W1

⊕W2

,

is a unitary operator on W.

Therefore, (V1, V2, V1V2) on H and (MΦ1 ,MΦ2 ,MWz ) on H2

W(D) are uni-

tarily equivalent, where W is the wandering subspace for V = V1V2.


4. Unitary invariants

In this short section we present a complete set of joint unitary invariants

for pure pairs of commuting isometries. Recall that two commuting pairs

(T1, T2) and (T1, T2) on H and H, respectively, are said to be (jointly) uni-

tarily equivalent if there exists a unitary operator U : H → H such that

UTj = TjU for all j = 1, 2.

First we note that, by virtue of Theorem 2.9 of [BDF], the orthogo-

nal projection PW2 and the unitary operator U on W , as in Theorem 3.3,

form a complete set of (joint) unitary invariants of pure pairs of commuting

isometries. More speci�cally: Let (V1, V2) and (V1, V2) be two pure pairs of

commuting isometries on H and H, respectively. Let Wj be the wandering

subspace for Vj, j = 1, 2. Then (V1, V2) and (V1, V2) are unitarily equivalent

if and only if

(

[V2|W1 0

0 V ∗1 |V1W2

], PW2) and (

[V2|W1

0

0 V ∗1 |V1W2

], PW2

)

are unitarily equivalent.

In addition to the above, the following unitary invariants are also explicit.

The proof is an easy consequence of Theorem 3.2. Here we will make use of

the identi�cations of A on H2W(D) and AMz on H

2W(D) with IH2(D) ⊗A on

H2(D)⊗W and Mz⊗A on H2(D)⊗W , respectively, where A ∈ B(W) (see

Section 2).

Theorem 4.1. Let (V1, V2) and (V1, V2) be two pure pairs of commuting

isometries on H and H, respectively. Then (V1, V2) and (V1, V2) are unitarily

equivalent if and only if (V1|W2 , V∗

2 |V2W1) and (V1|W2, V ∗2 |V2W1

) are unitarily

equivalent.

Proof. Let (MΦ1 ,MΦ2) and (MΦ1,MΦ2

) be the BCL pairs corresponding to

(V1, V2) and (V1, V2), respectively, as in Theorem 3.2. Let C1 = V1|W2 and

C2 = V ∗2 |V2W1 be the coe�cients of Φ1. Similarly, let C1 and C2 be the

coe�cients of Φ1.

Now let Z :W → W be a unitary such that ZCj = CjZ, j = 1, 2. Then

MΦ1 = IH2(D) ⊗ C1 +Mz ⊗ C2

= IH2(D) ⊗ Z∗C1Z +Mz ⊗ Z∗C2Z

= (IH2(D) ⊗ Z∗)(IH2(D) ⊗ C1 +Mz ⊗ C2)(IH2(D) ⊗ Z)

= (IH2(D) ⊗ Z∗)MΦ1(IH2(D) ⊗ Z).


Because MΦ2 = MzM∗Φ1

and MΦ2= MzM

∗Φ1, it follows that (MΦ1 ,MΦ2)

and (MΦ1,MΦ2

) are unitarily equivalent, that is, (V1, V2) and (V1, V2) are

unitarily equivalent.

To prove the necessary part, let (MΦ1 ,MΦ2) and (MΦ1,MΦ2

) are unitarily

equivalent. Then there exists a unitary operator X : H2W(D) → H2

W(D)

[RR] such that

XMΦj= MΦj

X (j = 1, 2).

Since

XMWz = XMΦ1MΦ2 = MΦ1

XX∗MΦ2X = MΦ1

MΦ2X = MW

z X,

there exists a unitary operator Z :W → W such that

X = IH2(D) ⊗ Z.

This and XMΦ1 = MΦ1X implies that

(IH2(D)⊗Z)(IH2(D)⊗C1 +Mz ⊗C2) = (IH2(D)⊗ C1 +Mz ⊗ C2)(IH2(D)⊗Z).

Hence (C1, C2) and (C1, C2) are unitarily equivalent. This completes the

proof of the theorem.

Observe that the set of joint unitary invariants {V1|W2 , V∗

2 |V2W1}, asabove, is associated with the coe�cients of Φ1 of the BCL pair (MΦ1 ,MΦ2)

corresponding to (V1, V2). Clearly, by duality, a similar statement holds for

the coe�cients of Φ2 as well: {V2|W1 , V∗

1 |V1W2} is a complete set of joint

unitary invariants for pure pairs of commuting isometries.

5. Pure isometries

In this section we will analyze pairs of commuting isometries (V1, V2) such

that either V1 or V2 is a pure isometry, or both V1 and V2 are pure isometries.

We begin with a concrete example which illustrates this particular class and

also exhibits its complex structure.

Recall that the Hardy space H2(D2) over the bidisc D2 is the Hilbert

space of all analytic functions on the bidisc D2 with square summable Tay-

lor coe�cients (see Rudin [R]). Let Mzj on H2(D2) be the multiplication

operator by the coordinate function zj, j = 1, 2. Note that (Mz1 ,Mz2) on

H2(D2) can be identi�ed with (Mz⊗IH2(D), IH2(D)⊗Mz) on H2(D)⊗H2(D),

and consequently, (Mz1 ,Mz2) on H2(D2) is a pair of doubly commuting (that

is, M∗z1Mz2 = Mz2M

∗z1) pure isometries.

Now let S be a joint (Mz1 ,Mz2)-invariant closed subspace of H2(D2), that

is, MzjS ⊆ S. SetVj = Mzj |S (j = 1, 2).


It follows immediately that Vj is a pure isometry and V1V2 = V2V1, and

hence (V1, V2) is a pair of commuting pure isometries on S.If we assume, in addition, that (V1, V2) is doubly commuting (that is, V ∗1 V2 =

V2V∗

1 ), then it follows that (V1, V2) on S and (Mz1 ,Mz2) on H2(D2) are uni-

tarily equivalent. See Slocinski [S] for more details. In general, however, the

classi�cation of pairs of commuting isometries, up to unitary equivalence,

is complicated and very little seems to be known. For instance, see Rudin

[R] for a list of pathological examples (also see Qin and Yang [QY]).

We now turn our attention to the general problem. Let (V1, V2) be a

pair of commuting isometries on H, and let V1 be a pure isometry. Then, in

particular, V = V1V2 is a pure isometry, and hence (V1, V2) is a pure pair of

commuting isometries. Since V1V2 = V2V1, by Theorem 2.2, it follows that

(5.1) ΠV1V2 = MΘV2ΠV1 ,

where ΘV2 ∈ H∞B(W1)(D) is an inner multiplier and

(5.2) ΘV2(z) = PW1(IH − zV ∗1 )−1V2|W1 (z ∈ D).

Let (MΦ1 ,MΦ2) be the BCL pair (see Theorem 3.3) corresponding to (V1, V2),

that is, ΠV Vi = MΦiΠV for all i = 1, 2. Set

Π1 = ΠV1Π∗V .

Then Π1 : H2W(D) → H2

W1(D) is a unitary operator such that Π1MΦ1 =

MW1z Π1 and Π1MΦ2 = MΘV2

Π1. Therefore, we have the following commu-

tative diagram:

H ΠV//

ΠV1 ##

H2W(D)

Π1

��

H2W1

(D)

where (MΦ1 ,MΦ2) on H2W(D) and (MW1

z ,MΘV2) on H2

W1(D) are the repre-

sentations of (V1, V2) on H.We now proceed to settle the non-trivial part of this consideration: An

analytic description of the unitary map Π1. To this end, observe �rst that

since ΠV1V1 = MW1z ΠV1 , (5.1) gives

ΠV1V = MW1z MΘV2

ΠV1 .

Then

Π1MWz = ΠV1VΠ∗V = MW1

z MΘV2ΠV1Π

∗V ,

that is,

(5.3) Π1MWz = (MW1

z MΘV2)Π1.


Let η ∈ W . By Equation (2.1) we can write η =∞∑

m=0

V m1 PW1V

∗m1 η. Therefore

(ΠV1η)(w) = (∞∑

m=0

ΠV1Vm

1 PW1V∗m

1 η)(w)

= (∞∑

m=0

Mmz PW1V

∗m1 η)(w)

=∞∑

m=0

wm(PW1V∗m

1 η),

which yields

Π1η = ΠV1Π∗V η = ΠV1η =

∞∑m=0

zm(PW1V∗m

1 η),

that is

Π1η = PW1 [IH + z(IH − zV ∗1 )−1V ∗1 ]η,

for all η ∈ W . It now follows from (5.3) that

Π1(zmη) = (zΘV2(z))mPW1 [IH + z(IH − zV ∗1 )−1V ∗1 ]η,

for all m ≥ 0, and so, by S(·, w)η =∞∑

m=0

zmwmη, it follows that

Π1(S(·, w)η) = Π1(∞∑

m=0

zmwmη)

= (IW1 − wzΘV2(z))−1PW1 [IH + z(IH − zV ∗1 )−1V ∗1 ]η,

for all w ∈ D and η ∈ W . Finally, from Π∗1MW1z = MΦ1Π

∗1 and Π∗1η1 = η1

for all η1 ∈ W1, it follows that Π∗1(zmη1) = MmΦ1η1 for all m ≥ 0, and hence

Π∗1(S(·, w)η1) = (IW − Φ1(z)w)−1η1,

for all w ∈ D and η1 ∈ W1.

We summarize the above observations in the following theorem.

Theorem 5.1. Let (V1, V2) be a pair of commuting isometries on H. Leti, j ∈ {1, 2} and i 6= j. If Vi is a pure isometry, then

Πi = ΠViΠ∗V ∈ B(H2

W(D), H2Wi

(D)),

is a unitary operator,

ΠiMWz = MzΘVj

Πi, Π∗iMWiz = MΦi

Π∗i ,

and

Πi(S(·, w)η) = (IWi− wzΘVj

(z))−1PWi[IH + z(I − zV ∗i )−1V ∗i ]η,


for all w ∈ D and η ∈ W, where

ΘVj(z) = PWi

(IH − zV ∗i )−1Vj|Wi

for all z ∈ D. Moreover

Π∗i (S(·, w)ηi) = (IW − Φi(z)w)−1ηi,

for all w ∈ D and ηi ∈ Wi.

Note that the inner multipliers ΘVi∈ H∞B(Wj)(D) above satisfy the fol-

lowing equalities:

ΠVjVi = MΘVi

ΠVj.

Now let (V1, V2) be a pair of commuting isometries such that both V1 and

V2 are pure isometries. The above result leads to an analytic representation

of such pairs.

Corollary 5.2. Let (V1, V2) be a pair of commuting pure isometries on a

Hilbert space H. If (MΦ1 ,MΦ2) is the BCL representation corresponding to

(V1, V2), then MΦ1 and MΦ2 are pure isometries,

Π1MΦ2 = MΘV2Π1, Π2MΦ1 = MΘV1

Π2,

Π = Π2Π∗1 : H2W1

(D)→ H2W2

(D) is a unitary operator, and

ΠMW1z = MΘV1

Π and ΠMΘV2= MW2

z Π.

Moreover, for each w ∈ D and ηj ∈ Wj, j = 1, 2,

Π(S(·, w)η1) = (IW2 − wΘV1(z))−1PW2(IH − zV ∗2 )−1η1,

and

Π∗(S(·, w)η2) = (IW1 − wΘV2(z))−1PW1(IH − zV ∗1 )−1η2.

Proof. A repeated application of Theorem 5.1 yields

Π1MΦ2 = Π1M∗Φ1

(MΦ1MΦ2)

= Π1M∗Φ1MW

z

= (MW1z )∗Π1M

Wz

= (MW1z )∗MzΘV2

Π1,

that is, Π1MΦ2 = MΘV2Π1 and similarly Π2MΦ1 = MΘV1

Π2. For η1 ∈ W1,

we have ΠV2η1 = PW2(IH − zV ∗2 )−1η1. Since Π∗1η1 = η1 and Π∗V η1 = η1, it

follows that

Πη1 = Π2η1 = ΠV2Π∗V η1 = ΠV2η1,

that is Πη1 = PW2(IH−zV ∗2 )−1η1. Now using the identity Π(zη1) = MΘV1Πη1,

we have

Π(zmη1) = ΘV1(z)mPW2(IH − zV ∗2 )−1η1,


for all m ≥ 0 and η1 ∈ W1. Finally S(·, w)η1 =∞∑

m=0

wmzmη1 gives

Π(S(·, w)η1) = (IW2 − wΘV1(z))−1PW2(IH − zV ∗2 )−1η1.

The �nal equality of the corollary follows from the equality

Π∗(zmη2) = ΘV2(z)m(Π∗η2) = ΘV2(z)mPW1(IH − zV ∗1 )−1η2,

for all m ≥ 0 and η2 ∈ W2. This concludes the proof.

In the �nal section, we will connect the analytic descriptions of Π1 and

Π2 as in Theorem 5.1 with the classical notion of the Sz.-Nagy and Foias

characteristic functions of contractions on Hilbert spaces [NF].

6. Defect Operators

Throughout this section, we will mostly work on general (not necessarily

pure) pairs of commuting isometries. Let (V1, V2) be a pair of commuting

isometries on a Hilbert space H. The defect operator C(V1, V2) of (V1, V2)

(cf. [HQY]) is de�ned as the self-adjoint operator

C(V1, V2) = I − V1V1∗ − V2V2

∗ + V1V2V1∗V2∗.

Recall from Section 3 that given a pair of commuting isometries (V1, V2),

we write V = V1V2, and denote by

Wj =W(Vj) = kerV ∗j = H VjH,

the wandering subspace for Vj, j = 1, 2. The wandering subspace for V is

denoted by W . Finally, we recall that (see Lemma 3.1) W =W1 ⊕ V1W2 =

V2W1 ⊕W2. This readily implies

(6.1) PW = PW1 ⊕ PV1W2 = PV2W1 ⊕ PW2 .

The following lemma is well known to the experts, but for the sake of

completeness we provide a proof of the statement.

Lemma 6.1. Let (V1, V2) be a commuting pair of isometries on H. ThenHs(V ) and Hu(V ) are Vj-reducing subspaces,

Hs(Vj) ⊆ Hs(V ), and Hu(Vj) ⊇ Hu(V ),

for all j = 1, 2.

Proof. For the �rst part we only need to prove that Hs(V ) is a V1-reducing

subspace. Note that since (see Lemma 3.1) V1W ⊆W⊕VW , it follows that

V1VmW ⊆ V m(W ⊕ VW) ⊆ Hs(V ),


for all m ≥ 0. This clearly implies that V1Hs(V ) ⊆ Hs(V ). On the other

hand, since V ∗1W =W2 ⊆ W and

V ∗1 VmW = V m−1(V2W) ⊆ V m−1(W ⊕ VW),

it follows that V ∗1 Hs(V ) ⊆ Hs(V ). To prove the second part of the state-

ment, it is enough to observe that

V mH = V m1 (V m

2 H) = V m2 (V m

1 H) ⊆ V m1 H, V m

2 H,

for all m ≥ 0, and as n→∞

V ∗n1 h→ 0, or V ∗n2 h→ 0⇒ V ∗nh→ 0,

for any h ∈ H. This concludes the proof of the lemma.

The following characterizations of doubly commuting isometries will prove

important in the sequel.

Lemma 6.2. Let (V1, V2) be a pair of commuting isometries on a Hilbert

space H. Then the following are equivalent:

(i) (V1, V2) is doubly commuting.

(ii) V2W1 ⊆ W1.

(iii) V1W2 ⊆ W2.

Proof. Since (i) implies (ii) and (iii), by symmetry we only need to show

that (ii) implies (i). Let V2W1 ⊆ W1. Let H = Hs(V ) ⊕ Hu(V ) be the

Wold-von Neumann orthogonal decomposition of V (see Theorem 2.1).

Then Hs(V ) and Hu(V ) are joint (V1, V2)-reducing subspaces, and the pair

(V1|Hu(V ), V2|Hu(V )) on Hu is doubly commuting, because Vj|Hu(V ), j = 1, 2,

are unitary operators, by Lemma 6.1. Now it only remains to prove that

V ∗1 V2 = V2V∗

1 on Hs(V ). Since

(V ∗1 V2 − V2V∗

1 )V m = V ∗1 VmV2 − V2V

∗1 V

m = V m−1V 22 − V 2

2 Vm−1 = 0,

it follows that V ∗1 V2 − V2V∗

1 = 0 on V mW for all m ≥ 1. In order to

complete the proof we must show that V ∗1 V2 = V2V∗

1 on W . To this end, let

η = η1 ⊕ V1η2 ∈ W for some η1 ∈ W1 and η2 ∈ W2. Then

V ∗1 V2(η1 ⊕ V1η2) = V ∗1 V2η1 + V ∗1 V2V1η2 = V2η2,

as V2W1 ⊆ W1, and on the other hand

V2V∗

1 (η1 ⊕ V1η2) = V2V∗

1 η1 + V2V∗

1 V1η2 = V2η2.

This completes the proof.

The key of our geometric approach is the following simple representation

of defect operators.


Lemma 6.3.

C(V1, V2) = PW1 − PV2W1 = PW2 − PV1W2 .

Proof. The result readily follows from (6.1) and

C(V1, V2) = (I − V1V1∗) + (I − V2V2

∗)− (I − V V ∗)

= PW1 + PW2 − PW .

The �nal ingredient to our analysis is the fringe operator F2. The notion

of fringe operators plays a signi�cant role in the study of joint shift-invariant

closed subspaces of the Hardy space over D2 (see the discussion at the

beginning of Section 5). Given a pair of commuting isometries (V1, V2) on

H, the fringe operators F1 ∈ B(W2) and F2 ∈ B(W1) are de�ned by

Fj = PWiVj|Wi

(i 6= j).

Of particular interest to us are the isometric fringe operators. Note that

F ∗2F2 = PW1V∗

2 PW1V2|W1 .

Lemma 6.4. The fringe operator F2 on W1 is an isometry if and only if

V2W1 ⊆ W1.

Proof. As IW1 − F ∗2F2 = IW1 − PW1V∗

2 PW1V2|W1 , (6.1) implies that

IW1 − F ∗2F2 = PW1V∗

2 PV1W2V2|W1 .

Then F ∗2F2 = IW1 if and only if PV1W2V2|W1 = 0, or, equivalently, if and

only if V2W1 ⊥ V1W2 =W⊥1 , by Lemma 3.1. This completes the proof.

Therefore, the fringe operator F2 is an isometry if and only if the pair

(V1, V2) is doubly commuting.

We are now ready to formulate a generalization of Theorem 3.4 in [HQY]

by He, Qin and Yang. Here we do not assume that (V1, V2) is pure.

Theorem 6.5. Let (V1, V2) be a pair of commuting isometries on H. Thenthe following are equivalent:

(a) C(V1, V2) ≥ 0.

(b) V2W1 ⊆ W1.

(c) (V1, V2) is doubly commuting.

(d) C(V1, V2) is a projection.

(e) The fringe operator F2 is an isometry.


Proof. The equivalences of (a) and (b), (b) and (c), and (b) and (e) are given

in Lemma 6.3, Lemma 6.2 and Lemma 6.4, respectively. The implication (c)

implies (d) follows from

C(V1, V2) = PW1PW2 = PW2PW1 .

Clearly (d) implies (a). This completes the proof.

We now prove that for a large class of pairs of commuting isometries

negative defect operator always implies the zero defect operator.

Theorem 6.6. Let (V1, V2) be a pair of commuting isometries on H. Sup-pose that V1 or V2 is pure. Then C(V1, V2) ≤ 0 if and only if C(V1, V2) = 0.

Proof. With out loss of generality assume that V2 is pure. If C(V1, V2) ≤ 0,

then by Lemma 6.3, we have PW1 ≤ PV2W1 , or, equivalently

W1 ⊆ V2W1,

and hence

W1 ⊆ V2mW1 ⊆ V2

mH,

for all m ≥ 0. Therefore

W1 =∞∩

m=0V2

mW1 ⊆∞∩

m=0V2

mH = {0},

as V2 is pure. Hence W1 = {0} and V2W1 = {0}. This gives C(V1, V2) =

PW1 − PV2W1 = 0.

The same conclusion holds if we allow dim Wj <∞ for some j ∈ {1, 2}.

Theorem 6.7. Let (V1, V2) be a pair of commuting isometries on H. Sup-pose that dim Wj <∞ for some j ∈ {1, 2}. Then C(V1, V2) ≤ 0 if and only

if C(V1, V2) = 0.

Proof. We may suppose that dim W1 <∞. Let C(V1, V2) ≤ 0. Since W1 ⊆V2W1 and V2 is an isometry, it follows that

W1 = V2W1.

Hence C(V1, V2) = PW1 − PV2W1 = 0. This completes the prove.

The same conclusion also holds for positive defect operators.


7. Concluding Remarks

As pointed out in the introduction, a general theory for pairs of com-

muting isometries is mostly unknown and unexplored (however, see Popovici

[P]). In comparison, we would like to add that a great deal is known about

the structure of pairs (and even of n-tuples) of commuting isometries with

�nite rank defect operators (see [BKS], [BKPS1], [BKPS2]). A complete

classi�cation result is also known for n-tuples of doubly commuting isome-

tries (cf. [GS], [S], [JS]). It is now natural to ask whether the present results

for pure pairs of commuting isometries can be extended to arbitrary pairs

of commuting isometries (see [D], [GG] and [GS] for closely related results).

Another relevant question is to analyze the joint shift invariant subspaces

of the Hardy space over the unit bidisc [ACD] from our analytic and geo-

metric point of views. More detailed discussion on these issues will be given

in forthcoming papers.

Also we point out that some of the results of this paper can be extended

to n-tuples of commuting isometries and will be discussed in a future paper.

We conclude this paper by inspecting a connection between the Sz.-Nagy

and Foias characteristic functions of contractions on Hilbert spaces [NF] and

the analytic representations of Π1 and Π2 as described in Theorem 5.1.

Let T be a contraction on a Hilbert space H. The defect operators of T ,

denoted by DT ∗ and DT , are de�ned by

DT ∗ = (I − TT ∗)1/2, DT = (I − T ∗T )1/2.

The defect spaces, denoted by DT ∗ and DT , are the closure of the ranges of

DT ∗ and DT , respectively. The characteristic function [NF] of the contrac-

tion T is de�ned by

θT (z) = [−T + zDT ∗(I − zT ∗)−1DT ]|DT(z ∈ D).

It follows that θT ∈ H∞B(DT ,DT∗ )(D) [NF]. The characteristic function is a

complete unitary invariant for the class of completely non-unitary contrac-

tions. This function is also closely related to the Beurling-Lax-Halmos inner

functions for shift invariant subspaces of vector-valued Hardy spaces. For

a more detailed discussion of the theory and applications of characteristic

functions we refer to the monograph by Sz.-Nagy and Foias [NF].


Now let us return to the study of pairs of commuting isometries. Let (V1, V2)

be a pair of commuting isometries on H. We compute

PW1 [IH + z(IH − zV ∗1 )−1V ∗1 ]|W = [PW1 + zPW1(IH − zV ∗1 )−1V ∗1 ]|W= [IH − V1V

∗1 + zPW1(IH − zV ∗1 )−1V ∗1 ]|W

= IW + [−V1 + zPW1(IH − zV ∗1 )−1]V ∗1 |W .

Since V ∗1W =W2, it follows that

[−V1+zPW1(IH−zV ∗1 )−1]V ∗1 |W = [−V1+zDV ∗1(IH−zV ∗1 )−1DV ∗2

]|DV ∗2(V ∗1 |W).

Therefore, setting

(7.1) θV1,V2(z) = [−V1 + zDV ∗1(IH − zV ∗1 )−1DV ∗2

]|DV ∗2,

for z ∈ D, we have

PW1 [IH + z(IH − zV ∗1 )−1V ∗1 ]|W = IW + θV1,V2(z)V ∗1 |W ,

for all z ∈ D. Therefore, if V1 is a pure isometry, then the formula for Π1 in

Theorem 5.1(i) can be expressed as

Π1(S(·, w)η) = (IW1 − wΘV2(z))−1PW1 [IW + θV1,V2(z)V ∗1 |W ]η.

for all w ∈ D and η ∈ W . Similarly, if V2 is a pure isometry, then the formula

for Π2 in Theorem 5.1 (ii) can be expressed as

Π2(S(·, w)η) = (IW2 − wΘV1(z))−1PW2 [IW + θV2,V1(z)V ∗2 |W ]η,

for all w ∈ D and η ∈ W , where

(7.2) θV2,V1(z) = [−V2 + zDV ∗2(IH − zV ∗2 )−1DV ∗1

]|DV ∗1,

for all z ∈ D.It is easy to see that θVi,Vj

(z) ∈ B(Wj,W) for all z ∈ D and i 6= j.

Note that since the defect operator DVj= 0, the characteristic function

θVjof Vj, j = 1, 2, is the zero function. From this point of view, it is

expected that the pair of analytic invariants {θVi,Vj: i 6= j} will provide

more information about pairs of commuting isometries.

Subsequent theory for pairs of commuting contractions and a more detailed

connection between pairs of commuting pure isometries (V1, V2) and the

analytic invariants {θVi,Vj: i 6= j} as de�ned in (7.1) and (7.2) will be

exhibited in more details in future occasion.


Acknowledgements. The authors are grateful to the anonymous review-

ers for their critical and constructive reviews and suggestions that have

substantially improved the manuscript. The �rst author's research work is

supported by NBHM Post Doctoral Fellowship No. 2/40(50)/2015/ R & D -

II/11569. The research of the second author was supported in part by (1) Na-

tional Board of Higher Mathematics (NBHM), India, grant NBHM/R.P.64/2014,

and (2) Mathematical Research Impact Centric Support (MATRICS) grant,

File No : MTR/2017/000522, by the Science and Engineering Research

Board (SERB), Department of Science & Technology (DST), Government

of India.

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Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile,

Mysore Road, Bangalore, 560059, India

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